While researching models for a recent Christmas Origami workshop I came across a pair of beautiful geometric gift bows; the perfect handmade touch for your gift wrapping this year.

The first bow (above middle) is the easiest. I used these high quality, printable instructions from the excellent Origami-fun website: Origami gift bow.

Alternatively, these video instructions from 1000 Crafts are clear and easy to follow.

The second gift bow is another 8-pointed star. This one has a slightly fiddly fold halfway through, but it is well explained in this video by Tavin's Origami Instructions, and has a stunning end result.

Each bow turns out to be about a quarter the size (area-wise) of the original sheet of paper.

So all that's left to do is to choose your gift-wrap colour-scheme :) I wrapped with brown kraft paper and found some beautiful matching mandala-themed paper that reminded me a little of snowflakes. Happy wrapping!

Last year I made my own Spirograph Christmas cards and I so enjoyed the process that this year I've decided to make my own mathsy cards again.

Recently I've been learning to draw a lot of Islamic geometric designs using compass and straight-edge, and I fancied using the same tools to create a Christmas card. I came up with the idea of constructing an accurate 5-pointed star (pentagram) and leaving the construction lines behind as part of the design.

Here's how they turned out :o)

I'm really happy with them, and they don't take too long to make if you get a little production line going. I thought I'd share the instructions so that you can make them too. I've also made a 'Construct a five-pointed star' powerpoint if you'd like to make them with your students as a 'Christmaths' activity.

Materials

You'll need a pair of compasses, a nice sharp pencil, a ruler and some coloured pens, pencils or paints. Metallic pens are great for getting a shimmery gold or silver outline. You'll also need some card to fold and construct your design on. Any sturdy 160-250 gsm card will do, but I used some pre-folded blank cards which can be bought cheaply in most stationery or craft shops.

Instructions

STEP 1 Measure to find the centre of the front of your card and mark this faintly with your pencil. Draw a horizontal line through this centre, from side to side.

STEP 2 Place your compass point on your marked centre and draw a starting circle that leaves at least two or three centimetres of card around its edge. You need the space around the outside of your circle for some construction lines in the next step.

STEP 4 Outline your star in marker pen, or metallic pen for extra shimmeriness. Then colour it in. I used metallic watercolour paints for mine.

STEP 5 Finally, if you wish, you can outline your star again with a fine black pen. I achieved a 'interleaved' effect on my stars by outlining both sides of my original metallic line, going alternatively 'over and under' each line I met as I travelled around the star (see bottom right photo below).

If you have lots of cards to draw, then it's easiest to set up a production line. Gather your blank cards together in a pile and draw one step of the construction process at a time on every card. That is, draw all your starting horizontal lines on every card first, then all your starting circles, then all your perpendicular bisectors and so on. This helps you avoid having to repeatedy open and close your compass.

What other Christmassy things could you make with your stars? Do let me know in the comments below.

The Hexastix sculpture has been on my to-do list since I first saw it on Twitter over a year ago. And yesterday I made one! And, well, the construction of this slightly bonkers mathematical structure made from 72 pencils, first created by George W. Hart, was so immensely satisfying that I've been inspired to blog to encourage EVERYONE to make their own!

Now life's too short to make your own set of instructions when there are perfectly good instructions already out there, so I'll just share a few tips that I learnt from making some rookie errors on my first attempt, and then link to the excellent instructions that I used.

But before I begin, here's my finished Hexastix! I think I'd describe it as a 3D structure made from four intersecting hexagonal 'prisms'...

Materials

Basically, you need 72 hexagonal-cross-sectioned pencils (with maybe a few spares) and a pack of elastic bands. You could use wooden skewers or toothpicks instead of pencils, but you'd miss out on the ultra-pleasing hexagon-shaped holes that emerge when using the pencils. I chose to use these plain wooden pencils and uncoloured elastic bands, as I wanted a 'natural' look, but TIP 1 is that colour is a BIG HELP when constructing this sculpture, so you may choose to use four different colours of pencil, or four different colours of elastic band. I resorted to taping a strip of coloured paper around one pencil in each of the four sets (below right) to help with the construction process, after getting in a complete muddle on my first attempt!

And TIP 2: unsharpened pencils are a wise choice if you don't fancy stabbing yourself several times in the construction process!

Construction

I would advise first watching this entertaining video by Matt Parker (@Standupmaths), so that you have an overview of what it is you'll be doing before you actually get started on the building process.

For the actual Hexastix construction, however, I recommend the same set of photo instructions that Matt does, these by Alejandro Erickson.

TIP 3: after my false start, I labelled my pencils in the same colours as Alejandro uses, and then started by orienting my structure to line up with his photos each time, so that I could follow his instructions exactly. However, soon enough, as the little hexagonal gaps start emerging, you'll get the hang of the construction and will just go for it! Once you have your initial 'Level 1' set of 4 hexagonal prisms, you 'expand' them twice more to achieve your very own Level 3 Hexastix!

Level 1

Level 2

Level 3!

TIP 4: be aware that the elastic bands will deteriorate over time. To preserve them as long as possible, keep your Hexastix out of direct sunlight.